------------------------------------------------------------------------ -- The Agda standard library -- -- The basic code for equational reasoning with three relations: -- equality, strict ordering and non-strict ordering. ------------------------------------------------------------------------ -- -- See `Data.Nat.Properties` or `Relation.Binary.Reasoning.PartialOrder` -- for examples of how to instantiate this module. {-# OPTIONS --without-K --safe #-} open import Relation.Binary module Relation.Binary.Reasoning.Base.Triple {a ℓ₁ ℓ₂ ℓ₃} {A : Set a} {_≈_ : Rel A ℓ₁} {_≤_ : Rel A ℓ₂} {_<_ : Rel A ℓ₃} (isPreorder : IsPreorder _≈_ _≤_) (<-trans : Transitive _<_) (<-resp-≈ : _<_ Respects₂ _≈_) (<⇒≤ : _<_ ⇒ _≤_) (<-≤-trans : Trans _<_ _≤_ _<_) (≤-<-trans : Trans _≤_ _<_ _<_) where open import Data.Product using (proj₁; proj₂) open import Function.Base using (case_of_; id) open import Level using (Level; _⊔_; Lift; lift) open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; sym) open import Relation.Nullary using (Dec; yes; no) open import Relation.Nullary.Decidable using (True; toWitness) open IsPreorder isPreorder renaming ( reflexive to ≤-reflexive ; trans to ≤-trans ; ∼-resp-≈ to ≤-resp-≈ ) ------------------------------------------------------------------------ -- A datatype to abstract over the current relation infix 4 _IsRelatedTo_ data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where strict : (x<y : x < y) → x IsRelatedTo y nonstrict : (x≤y : x ≤ y) → x IsRelatedTo y equals : (x≈y : x ≈ y) → x IsRelatedTo y ------------------------------------------------------------------------ -- Types that are used to ensure that the final relation proved by the -- chain of reasoning can be converted into the required relation. data IsStrict {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where isStrict : ∀ x<y → IsStrict (strict x<y) IsStrict? : ∀ {x y} (x≲y : x IsRelatedTo y) → Dec (IsStrict x≲y) IsStrict? (strict x<y) = yes (isStrict x<y) IsStrict? (nonstrict _) = no λ() IsStrict? (equals _) = no λ() extractStrict : ∀ {x y} {x≲y : x IsRelatedTo y} → IsStrict x≲y → x < y extractStrict (isStrict x<y) = x<y data IsEquality {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where isEquality : ∀ x≈y → IsEquality (equals x≈y) IsEquality? : ∀ {x y} (x≲y : x IsRelatedTo y) → Dec (IsEquality x≲y) IsEquality? (strict _) = no λ() IsEquality? (nonstrict _) = no λ() IsEquality? (equals x≈y) = yes (isEquality x≈y) extractEquality : ∀ {x y} {x≲y : x IsRelatedTo y} → IsEquality x≲y → x ≈ y extractEquality (isEquality x≈y) = x≈y ------------------------------------------------------------------------ -- Reasoning combinators -- See `Relation.Binary.Reasoning.Base.Partial` for the design decisions -- behind these combinators. infix 1 begin_ begin-strict_ begin-equality_ infixr 2 step-< step-≤ step-≈ step-≈˘ step-≡ step-≡˘ _≡⟨⟩_ infix 3 _∎ -- Beginnings of various types of proofs begin_ : ∀ {x y} → x IsRelatedTo y → x ≤ y begin (strict x<y) = <⇒≤ x<y begin (nonstrict x≤y) = x≤y begin (equals x≈y) = ≤-reflexive x≈y begin-strict_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsStrict? r)} → x < y begin-strict_ r {s} = extractStrict (toWitness s) begin-equality_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsEquality? r)} → x ≈ y begin-equality_ r {s} = extractEquality (toWitness s) -- Step with the strict relation step-< : ∀ (x : A) {y z} → y IsRelatedTo z → x < y → x IsRelatedTo z step-< x (strict y<z) x<y = strict (<-trans x<y y<z) step-< x (nonstrict y≤z) x<y = strict (<-≤-trans x<y y≤z) step-< x (equals y≈z) x<y = strict (proj₁ <-resp-≈ y≈z x<y) -- Step with the non-strict relation step-≤ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≤ y → x IsRelatedTo z step-≤ x (strict y<z) x≤y = strict (≤-<-trans x≤y y<z) step-≤ x (nonstrict y≤z) x≤y = nonstrict (≤-trans x≤y y≤z) step-≤ x (equals y≈z) x≤y = nonstrict (proj₁ ≤-resp-≈ y≈z x≤y) -- Step with the setoid equality step-≈ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≈ y → x IsRelatedTo z step-≈ x (strict y<z) x≈y = strict (proj₂ <-resp-≈ (Eq.sym x≈y) y<z) step-≈ x (nonstrict y≤z) x≈y = nonstrict (proj₂ ≤-resp-≈ (Eq.sym x≈y) y≤z) step-≈ x (equals y≈z) x≈y = equals (Eq.trans x≈y y≈z) -- Flipped step with the setoid equality step-≈˘ : ∀ x {y z} → y IsRelatedTo z → y ≈ x → x IsRelatedTo z step-≈˘ x y∼z x≈y = step-≈ x y∼z (Eq.sym x≈y) -- Step with non-trivial propositional equality step-≡ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≡ y → x IsRelatedTo z step-≡ x (strict y<z) x≡y = strict (case x≡y of λ where refl → y<z) step-≡ x (nonstrict y≤z) x≡y = nonstrict (case x≡y of λ where refl → y≤z) step-≡ x (equals y≈z) x≡y = equals (case x≡y of λ where refl → y≈z) -- Flipped step with non-trivial propositional equality step-≡˘ : ∀ x {y z} → y IsRelatedTo z → y ≡ x → x IsRelatedTo z step-≡˘ x y∼z x≡y = step-≡ x y∼z (sym x≡y) -- Step with trivial propositional equality _≡⟨⟩_ : ∀ (x : A) {y} → x IsRelatedTo y → x IsRelatedTo y x ≡⟨⟩ x≲y = x≲y -- Termination step _∎ : ∀ x → x IsRelatedTo x x ∎ = equals Eq.refl -- Syntax declarations syntax step-< x y∼z x<y = x <⟨ x<y ⟩ y∼z syntax step-≤ x y∼z x≤y = x ≤⟨ x≤y ⟩ y∼z syntax step-≈ x y∼z x≈y = x ≈⟨ x≈y ⟩ y∼z syntax step-≈˘ x y∼z y≈x = x ≈˘⟨ y≈x ⟩ y∼z syntax step-≡ x y∼z x≡y = x ≡⟨ x≡y ⟩ y∼z syntax step-≡˘ x y∼z y≡x = x ≡˘⟨ y≡x ⟩ y∼z